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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Non random sequences

In this pdf, they introduce something called "non-random" sequences. What does that mean? Other than $X(\omega) = i, \;\forall i.$
music
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Asymptotic stability of Continuous Time Markov Chains (CTMCs)

The evolution of the state occupancy probabilities of a CTMC can be described by a linear system of differential equations of the form $\dot{\mathbf{x}} = A.\mathbf{x}$. The rate matrix A usually has a 0 zero eigenvalue. Therefore, A violates the…
user98568
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Regarding change of measure

Suppose a Gaussian process $\{B_t\}$, apparently $2^{B_t}$ is not a martingale. Can someone teach me how to change the measure so that $2^{B_t}$ is a martingale? Thanks.
user90846
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Stochastic integral of local martingale is local martingale provided it is bounded below

The following Theorem I have thought for a long time. I feel like there seems to be no simple solution for it. Does anyone have any ideas or hints for this? Let $H$ be a previsible process. $X$ be a local martingale. Suppose $H$ is integrable with…
Zorualyh
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Understanding the sample space and probability distribution of a Gaussian process

I'm trying to unpack the mathematical definition of a Gaussian process, by applying it to an example problem of modelling the height $h(t)$ of the tide at time $t$. Let's begin with two definitions of a Gaussian process: From Wikipedia: A Gaussian…
Ruvi Lecamwasam
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Canonical Filtration of a partial sum

I wonder if the canonical filtration of a partial sum of a discrete stochastic process is equal to the canonical filtration of the process it self e.g. Is ${\sigma}(X_1,\ldots,X_n)={\sigma}(X_1,X_1+X_2,\ldots,X_1+\ldots+X_n)$ where $(X_n)_{n\in…
MackeyTopology
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Coverage probability in a Spatial Boolean Process

Consider a Poisson Boolean Process (X,$\lambda$,1) where $\lambda>0$ is the Poisson intensity of a two dimensional Poisson process. The Boolean process is such that at the center of each Poisson point a square tile is centered with side 1. (Usually…
Aris
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Markov chain which at each transition either goes $+1$ with probability $p$ or $-1$ with probability $q$. $(q/p)^{S_n}, n \geq 1$ is a martingale.

Consider the Markov chain which at each transition either goes up $1$ with probability $p$ or down $1$ with probability $q = 1 - p$. Argue that $(q/p)^{S_n}, n \geq 1$ is a martingale. I tried to show $E[Z_{n+1}|Z_1,...,Z_n] = Z_n$ as…
Math_Day
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Is the expectation of the quadratic variation of two independent random process always 0?

My guess is this holds, for example, if $X(t) = c$ is constant, $Y(t) = W(t)$ is Brownian motion, then their quadratic variation is $\langle X, Y \rangle (t) = W(t)$ which has mean of 0; $X(t) = W_1(t)$ and $Y(t) = W_2(t)$ also work; $X(t) = W(t)$…
athos
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Question about canonical markov process with strong markov property.

I'm really stuck on this bit, maybe someone can help be along: Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel $\sigma$-algebra $\mathcal{E}$. Assume $t \mapsto…
BallzofFury
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Heaviside function as a càdlàg adapted process of finite variation

As written in Wikipedia, the Dirac delta function is the derivative of the Heaviside function $${\displaystyle \delta (x)={\frac {d}{dx}}H(x)}$$ Hence the Heaviside function can be considered to be the integral of the Dirac delta…
Lely
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Can you define a stochastic process via its marginals?

If you are given a finite fumber of r.v. say $X_1...X_n$ and you know say the conditional distribution of each one given the preceding ones (i.e with smaller index) values , and you also know the marginal say $X_1$ then there is a unique measure…
eternalstudent
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Solution of a particular Backward SDE

I have been reading about backward SDEs and their existence-uniqueness result. The results are available for equations of the form $$ -dY_t = f(t,\omega, Y_t,Z_t)\,dt- Z_t\,dW_t, \quad Y_T= \xi.$$ A solution is a pair $(Y_t,Z_t)$ such that $Z_t$ is…
Savannah
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Understanding the "reflection of paths" intuition behind the reflection equality for Brownian Motion

Let $W(t)$ be a Brownian Motion for $t\geq 0$, and $\tau_m$ be the first passing time of $W(t)$ above $m$ such that $\tau_m=\text{inf}\{t: B_t\geq m\}$. Then, according to Shreve's Stochastic Calculus for Finance II equation…
mesutkomiser
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How to show that this element is predictable

Suppose I have a process $Y_t$, which is predictable. We can assume that the filtration satisfies the usual condition. Then, let $A\in \mathcal{F}_t$. I was wondering about the following. Is $\mathbf1_AY_t$ still a predictable process? This seems…
user20869
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